THE GREAT ALGORIST integers: (#2 — t/2) = (x -f- y) (x — y]. But the same problem for #2 -f- y2 demands 'imaginaries': xz + y2 = (# + 2/V — 1) (x — y V — 1). Carrying this up a step in one of many possible ways open, we might seek to resolve #2 -f y2 + s2 into two factors of the first degree. Are the positives> negatives, and imaginaxies sufficient? Or must some new kind of 'number' be invented to solve the problem? The latter is the case. It was found that for the new 'numbers' necessary the rules of common algebra break down in one important particular: it is no longer true that the order in which 'numbers' are multiplied together is indifferent; that is, for the new numbers it is not true that a X & is equal to b x a. More will be said on this when we come to Hamilton. For the moment we note that the elementary alge- braic problem of factorization quickly leads us into regions where complex numbers are inadequate. How far can we go, what are the most general numbers possible, if we insist that for these numbers all the familiar laws of common algebra are to hold? It was proved in the latter part of the nineteenth century that the complex numbers x + iy, where xsy are real numbers and i — V — 1, are the most general for which common algebra is true. The real numbers, we recall, correspond to the distances measured along a fixed straight line in either direction (positive, negative) from a fixed point, and the graph of a function /(#), plotted as y — /(#), in Cartesian geometry, gives us a picture of a function y of a real variable x. The mathematicians of the seventeenth and eigh- teenth centuries imagined their functions as being of this kind. But if the common algebra and its extensions into the calculus which they applied to their functions are equally applicable to complex numbers, which include the real numbers as a very degenerate case, it was but natural that many of the things the early analysts found were less than half the whole story possible. In particular the integral calculus presented many inexplicable anomalies which were cleared up only when the field of operations was enlarged to its fullest possible extent and functions of complex variables were introduced by Gauss and Cauchy. The importance of elliptic functions hi all this vast and 867